Revealing low-loss dielectric near-field modes of hexagonal boron nitride by photoemission electron microscopy

Low-loss dielectric modes are important features and functional bases of fundamental optical components in on-chip optical devices. However, dielectric near-field modes are challenging to reveal with high spatiotemporal resolution and fast direct imaging. Herein, we present a method to address this issue by applying time-resolved photoemission electron microscopy to a low-dimensional wide-bandgap semiconductor, hexagonal boron nitride (hBN). Taking a low-loss dielectric planar waveguide as a fundamental structure, static vector near-field vortices with different topological charges and the spatiotemporal evolution of waveguide modes are directly revealed. With the lowest-order vortex structure, strong nanofocusing in real space is realized, while near-vertical photoemission in momentum space and narrow spread in energy space are simultaneously observed due to the atomically flat surface of hBN and the small photoemission horizon set by the limited photon energies. Our approach provides a strategy for the realization of flat photoemission emitters.


Supplementary Note 3 Discussion on the selective excitation of TE modes in thin planar waveguide
Here, we will demonstrate that when the hBN waveguide is thin enough (e.g. <100 nm), only the TE mode can be efficiently excited whereas the TM mode is negligible, so that we can use the TE mode of the waveguide to construct dielectric near-field modes. The discussions on the intensity of TE and TM modes with different hBN thicknesses are as follows. For the hBN thickness of 60 nm, the electric field amplitudes at the top surface (Z = −2 nm, that is 2 nm below the vacuum/hBN interface) and central plane (Z = −32 nm) of the hBN waveguide are shown in Supplementary Figure 3g. As we can see, the electric field intensity (|E| 2 ) of TE mode is much stronger than TM mode when the hBN is thin enough. Typically, at the thickness of 40 (or 60) nm, the electric field intensity ratios of TE/TM modes at the top surface and the central plane are about 7×10 3 (or 1×10 3 ) and 1.3×10 4  For the circular slits, strong near-field enhancement of the focusing spot is produced by the nearfield vortex excited with the circularly polarized beam. The size of the focusing spot achieves 58 nm for |E| 4 , apparently beyond the diffraction limit (Supplementary Figure 5a). Here, |E| 4 is used considering the two-photon process of photoemission in PEEM experiments. The near-field enhancement increases with the radius of the circular slit and the number of slits (circular Bragg grating). Noticeably, the near-field amplitude |E| increases linearly with the radius of the circular slit due to the low loss of dielectric material (Supplementary Figure 5b). The electric field intensity (|E| 2 ) can be enhanced over 10 3 times compared with the incident beam by using a circular Bragg grating with 10 slits and an initial radius of 30 (Supplementary Figure 5c). In contrast, the near-field intensity of SPP at the surface of the metal film is much weaker and decays fast due to the high loss of metal. In addition, the simulation results above are also applicable to the symmetrical waveguide, that is , which can be realized with free-standing hBN flakes in experiments. is also ~230 nm. The dot lattices can be observed for square slits excited with right-handed circular polarization, which support optical meron-like spin textures.

Supplementary Note 4 Two-photon photoemission process
For the excitation wavelength around 410 nm, two-photon photoemission process satisfies the condition of overcoming the work function of hBN, that is ∝ ∝ | | , where PE is the photoemission intensity, I is the local field intensity and |E| is the electric field amplitude. As

Supplementary Note 5 Discussions on the choice of conductive layer ITO and charging effect
It should be noted, a very thin ITO down to 10 nm is adopted in this work as conductive layer in order to avoid introducing absorption loss. ITO has much lower absorption loss than thin metallic film, such as gold and monolayer graphene. However, ITO itself still has a small absorption, therefore, the thickness down to 10 nm is preferred to maintain the low-loss hBN Simulated decay of waveguide mode in hBN along propagation direction with different ITO layer thicknesses: 0 nm, 10 nm and 50 nm. The hBN waveguide with 2 nm Au as conductive layer is also plotted for comparison. The waveguide mode is excited from a line slit on 60 nm hBN with 410 nm laser at normal incidence.
The low loss of dielectric material is a prerequisite for strong photoemission enhancement. The enhancement is achieved by focusing the waveguide mode excited from the ring slit edge. That's to say, the waveguide mode is coupled into the hBN slab from the slit edge and propagates toward the center to form a strong nanofocusing. Therefore, the loss during propagation will have a vital influence on the final focusing intensity. From the simulations, without considering loss, the focusing intensity increases with the ring diameter, because more light is collected by the slit edge with a larger circumference. And the focusing intensity can be further largely enhanced by using circular grating coupler. As for SPP supported by metal, large loss is expected, resulting in weak enhancement. And it's expected that the focusing intensity cannot be efficiently enhanced by increasing the diameter due to the propagation loss of SPP. As shown in Supplementary Figure 9, by using the same design, the intensity of focusing spot with hBN waveguide is much larger than that with gold film. The intensity |E| 2 at vacuum/hBN is ~12 times larger than that at vacuum/Au.
In addition, the maximum intensity of hBN waveguide is not at the surface, but inside hBN, as shown in Supplementary Figure 9a, which is ~23 times larger than that at vacuum/Au. It should be noted, for SPP, the ring slit (m=0) excited with LCP does not form a focusing point in the center, but a doughnut pattern. To form a focusing point, spiral slit (m=−1) should be adopted and we use the focusing point to compare the intensity. In addition, we can try to measure the loss from the decrease of photoemission intensity during propagation. As shown in Supplementary Figure 10a  In addition, as shown in Supplementary Figure 7b, the excitation density could have some influence on charging effect, because we observed that when the laser power was above 300 mW, the photoemission intensity became saturated. This saturation effect could be attributed to the limited charge transfer efficiency from hBN to ITO.
To note, the hBN used in this study is relatively small flakes (in-plane size <300 μm), rather than film that coving the whole surface, which could also be an advantage for transferring charges from hBN to underlying ITO. In addition, the hBN flake can be clearly imaged by PEEM without any blur. For example, the hBN flakes can also be clearly imaged by PEEM with Si substrate.
The careful selection of materials to balance the bandgap and surface charging is needed because bandgap and conductivity is contradictory to some extent, large bandgap generally means low conductivity. To construct optical devices in visible range, large bandgap is preferred. However, common optical materials such as Si3N4, LiNbO3 are not conductive, thus cannot be measured in PEEM. The materials with smaller bandgap such as Si, GaAs have sufficient conductivity, but they are not suitable for visible range due to large optical absorption. Therefore, the selection of materials is an important task. It could be a good idea to seek among novel materials. The van der Waals materials could be potential choices as they have special layered structure. And through the PEEM experiments, we found hBN is compatible with PEEM. It should be noted, it's hard to predict if the materials are suitable for PEEM measurements, the experimental attempts should be required.

Supplementary Note 6 PEEM images for near-field modes excited with right-handed circular and linear polarizations
For the Archimedean spiral slits with a series of geometrical charges (+m), the near-field vortex

Supplementary Note 7 Discussion on the optical spin textures carried by vector near-field vortex modes
The vector near-field vortex mode excited with a left-handed circularly polarized plane beam To better analyze the SAM texture of the near-field vortex, the SAM of the vortex is calculated where, ω is the angular frequency, ε and μ are the permittivity and permeability, and the asterisks denote complex conjugation. The SAM vectors at the top surface of hBN present a distinct Néeltype skyrmion-like spin texture (Supplementary Figure 13b,d), which is similar to that found in SPP vortex. From the center of the vortex to the periphery at the L-line (L2), the SAM vectors rotate continuously from upward to the radial, and finally to downward direction. The topological charge N of the SAM texture is calculated by integrating the charge density within the periphery 2,3 .

∬ •
Where, | | ⁄ is the unit spin vector. The obtained N is close to 1, confirming the skyrmion topological character.

Supplementary Note 9 Calculations for group velocity
The group velocity can be extracted from the moving of wave packet by the relationship: